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Acta Materialia 59 (2011) 3883–3894
www.elsevier.com/locate/actamat
Atomistic study of the buckling of gold nanowires
Pär A.T. Olsson a,⇑, Harold S. Park b
b
a
Division of Mechanics, Lund University, PO Box 118, SE-221 00 Lund, Sweden
Department of Mechanical Engineering, Boston University, Boston, MA 02215, USA
Received 25 November 2010; received in revised form 25 January 2011; accepted 6 March 2011
Available online 12 April 2011
Abstract
In this work, we present results from atomistic simulations of gold nanowires under axial compression, with a focus on examining the
effects of both axial and surface orientation effects on the buckling behavior. This was accomplished by using molecular statics simulations while considering three different crystallographic systems: h1 0 0i/{1 0 0}, h1 0 0i/{1 1 0} and h1 1 0i/{1 1 0}{1 0 0}, with aspect ratios
spanning from 20 to 50 and cross-sectional dimensions ranging from 2.45 to 5.91 nm. The simulations indicate that there is a deviation
from the inverse square length dependence of critical forces predicted from traditional linear elastic Bernoulli–Euler and Timoshenko
beam theories, where the nature of the deviation from the perfect inverse square length behavior differs for different crystallographic
systems. This variation is found to be strongly correlated to either stiffening or increased compliance of the tangential stiffness due to
the influence of nonlinear elasticity, which leads to normalized critical forces that decrease with decreasing aspect ratio for the h1 0 0i/
{1 0 0} and h1 0 0i/{1 1 0} systems, but increase with decreasing aspect ratio for the h1 1 0i/{1 1 0}{1 0 0} system. In contrast, it was found
that the critical strains are all lower than their bulk counterparts, and that the critical strains decrease with decreasing cross-sectional
dimensions; the lower strains may be an effect emanating from the presence of the surfaces, which are all more elastically compliant than
the bulk and thus give rise to a more compliant flexural rigidity.
Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Buckling; Nanowire; Molecular statics
1. Introduction
In recent years, extensive research efforts have been
devoted to increasing the understanding of the mechanical
properties of nanowires. The underlying reason for this is
because nanowires are potential candidates as components
in future nanoelectromechanical systems (NEMS) [1–3].
Furthermore, due to their nanometer dimensions, nanostructures tend to display physical properties that often differ markedly from that of macroscopic structures. The
basic reason for this is because surface atoms have a different bonding environment due to their reduced coordination
as compared to bulk atoms, which results in the enhancement of non-bulk physical properties with decreasing
⇑ Corresponding author. Tel.: +46 46 2223865; fax: +46 46 2224620.
E-mail addresses: par.olsson@mek.lth.se (P.A.T. Olsson), parkhs@
bu.edu (H.S. Park).
nanostructure size, or increasing surface area to volume
ratio [2,3].
Experimentally, the elastic properties of nanowires are
usually determined either through bending [4–10], tensile
[11,12] or dynamic resonant experiments [3,13–25], where
the elastic properties are extracted from the continuum
mechanical relations connecting the deflection, strain, force
or resonant properties with the Young’s modulus.
Recently, another mechanical testing technique, that of
buckling, has been utilized, where the elastic properties
are obtained by determining the critical force needed to
cause buckling instability of the nanowires in compression,
and subsequently relating the critical force to existing continuum relations from which the elastic properties are
extracted [26–29].
The elastic stability and behavior of nanowires under
compression have also become critical for other applications. For example, recent experiments on gold nanowires
with cross-sectional dimensions as small as 3 nm have
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2011.03.012
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shown that nanowires can be joined together through cold
welding. This occurs through the pressing together of two
nanowires, resulting in welding through atomic diffusion
at the interface of the nanowires [30]. For this application,
it is essential that the nanowires are not exposed to overly
large compressive forces, as the stability, and thus the
ability to cold weld the nanowires may be degraded.
Nanowires under the influence of axial strain have also
attracted increasing interest lately as one may mechanically tune the resonant frequencies using tensile or compressive strains. This is important as it opens up the
possibility of creating high-quality-factor micro- and
nanoresonators with variable eigenfrequency spectra suitable for a number of applications [23–25,31,32]. However,
if the compressive force or strain exceeds the critical force
or strain, then the nanowire may not vibrate controllably
due to the onset of buckling instability. Finally, recent
developments have paved the way for employment of
highly flexible and stretchable electronics [33,34], which
are now beginning to utilize wavy, buckled nanowires that
can undergo significant amounts of stretching before plastic deformation occurs. In this type of application, surface
effects on the nanowires and their buckling behavior and
properties may play a central role in the performance of
the devices.
These applications all demonstrate that it is critical to
investigate and understand not only how nanowires behave
under the influence of compressive loads, but also to study
how well critical loads and strains predicted from traditional continuum mechanics agree with those obtained
from direct atomistic modeling of the surface-dominated
nanowires.
Molecular dynamics and molecular statics (MS) have
been popular numerical tools for simulating mechanical
properties of nanowires and have revealed many interesting
features such as elastic size and crystallographic dependence, asymmetric yielding properties, nonlinear elastic
properties, shape memory, spontaneous phase transformations due to surface stress, pseudoelastic behavior, etc.
[31,32,35–54]. When considering buckling properties, most
atomistic simulations have shown that the critical strains
and forces are found to be significantly larger than what
can be predicted from fundamental Bernoulli–Euler buckling predictions [49–52]. Moreover, there have been reports
of the critical strain being dependent on the applied strain
increment size and it has been found that the stress–strain
and energy–strain curves may become discontinuous when
buckling occurs [47–54]. Strain increment size dependencies
and discontinuous energy–strain curves imply that the
transition between the straight beam to the buckled beam
does not necessarily follow the lowest energy path, which
complicates the extraction of the critical buckling force
and strain at which the straight-beam configuration
becomes unstable. We also note that the literature on the
buckling of nanowires is considerably less well-developed
than the extensive literature on the buckling of carbon
nanotubes.
The purpose of this paper is to study, using MS simulations, the effects of nanowire size, axial orientation and surface orientation on the buckling properties of gold
nanowires, and to evaluate how well the critical loads
and strains predicted from traditional continuum mechanics as well as recently developed surface elasticity theories
[55] agree with our simulation results. In addition to the
buckling studies, we detail a simple procedure that we have
utilized to address the issues regarding strain increment
dependencies and discontinuous energy–strain and stress–
strain curves that have been observed from previous atomistic studies of nanostructure buckling [47,51–54].
2. Preliminaries
For slender beams loaded in compression, the load-carrying ability and the geometry of the beam are highly
dependent on the magnitude of the compressive force.
For small loads, the geometry and the load-carrying ability
are not prone to undergo any noticeable changes and the
state is considered to be stable. However, increasing the
compressive force will eventually lead to a threshold value
at which the straight-beam equilibrium configuration
becomes unstable. This means that a small lateral force
or asymmetry will induce bowing deformations which will
not disappear despite removal of the lateral force, which
may result in loss of load-carrying ability for the beam.
In classical elasticity theory, the critical load for an ideal
doubly clamped Bernoulli–Euler beam is written as:
F cr ¼ 4p2 ðEIÞ
;
L2
ð1Þ
where L denotes the length of the beam and (EI) is the
flexural rigidity, which is given by:
Z
ðEIÞ ¼
Ez2 dA;
ð2Þ
A
where E denotes Young’s modulus and z is the coordinate
normal to the plane spanned by the wire axis and the crosssectional rotation axis, measured from the neutral axis. For
a homogeneous cross-section, the critical strain can be
written as:
cr ¼ 4p2 I
;
AL2
ð3Þ
where we have used the linear elastic relation between
stress and strain, which is valid for strains in the linear elastic regime, and A and I are the cross-sectional area and moment of inertia, respectively [56]. The minus signs in Eqs.
(1) and (3) indicate that the critical forces and strains are
compressive. We note that Eq. (3) demonstrates that, in
the macroscopic continuum beam theory where the crosssections are assumed to be homogeneous, the critical strain
is only a function of the beam geometry, and not of the
elastic properties.
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Fig. 1. Schematic view of the nanowire in the simulations.
Table 1
Cross-sectional sizes in the format h w in nm2 for the simulated
nanowires.
h1 0 0i/{1 0 0}
h1 0 0i/{1 1 0}
h1 1 0i/{1 1 0}{1 0 0}
2.45 2.45
3.26 3.26
4.08 4.08
5.71 5.71
2.45 2.45
3.32 3.32
4.18 4.18
5.91 5.91
3.32 3.47
4.18 4.28
5.91 5.92
3. Numerical details
3.1. Geometries
The considered nanowires are defect-free gold single
crystals with cross-sections as close to perfect squares as
possible. Three types of systems are considered, where
the orthogonal directions (x1,x2, x3) in Fig. 1 correspond
to the ([1 0 0], [0 1 0], [0 0 1]), ([1 0 0], [0 1 1], [0 1 1]) and
([1 1 0], [0 0 1], [110]) directions so that two different types
of free surfaces are considered, i.e. {1 0 0} and {1 1 0} surfaces, and two different types of axial directions are investigated, i.e. h1 0 0i and h1 1 0i. These systems will
subsequently be referred to as h1 0 0i/{1 0 0},h1 0 0i/{1 1 0}
and h1 1 0i/{1 1 0}{1 0 0} systems, respectively. The considered cross-sectional sizes can be seen in Table 1 for the different systems and we have considered aspect ratios
ranging from 20 to 50.
The atomistic systems are divided into three different
regions, A, B and C, as illustrated in Fig. 1. Regions A and
C are boundary regions whose main purpose are to mimic
doubly clamped boundary conditions. Region B consists of
free atoms which constitute the actual nanowire, hence the
length of nanowire is the same as the measured length of
region B. The lengths of regions A and C differ for different
simulations but they were chosen to be no less than four lattice constants long each, which is sufficiently long so that
region B is not influenced by the free end surfaces.
3.2. MS simulations
MS simulations are performed for the numerical part of
this work and the simulations are conducted as follows.
First we let the nanowire relax to a potential energy minimum using the FIRE algorithm [57], where throughout this
initial minimization all the atoms are allowed to move
freely without any constraints, except for the atoms in
region A in Fig. 1 which are fixed throughout the entire
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simulation. This relaxation step was performed such that
the atoms could find new equilibrium positions in response
to the surface stress that acts on the nanowire; the new
equilibrium configuration corresponds to a reduction in
the nanowire length of the order of 1–3% depending on
the cross-sectional size, surrounding surfaces and axial
orientation.
Once this initially relaxed stress-free state of equilibrium
was found, the atoms in region C were translated in the
negative x1 direction and held fixed, while the free atoms
in B were allowed to move to the potential-energy-minimizing positions using the FIRE algorithm. This displacementcontrolled compressive straining procedure was repeated
several times so that the stress measured in region B and
the reaction force acting on region C could be measured
as a function of the strain. In all the simulations we consistently measure the strains from the initially relaxed state
where the nanowire length has been reduced due to the surface stresses, so that the zero strain state corresponds to
that where the axial stress has been balanced. This means
that the ideal interplanar distance along the wire axis, in
general, does not correspond to the zero strain state. The
convergence criterion was chosen so that convergence
was assumed to be met when the root-mean-square force
of the system was less than 1011 eV/Å. The interatomic
interaction is modeled through an embedded atom method
(EAM) potential, fitted to gold properties [58]. This potential describes the elastic properties of gold very well, which
is of great importance in this study.
Importantly, the above numerical scheme alone is not
sufficient for calculating the critical force and strain as
the amount of force and strain at which the nanowires
buckle depends on the displacement increment size. Moreover, there is no guarantee that the nanowire will buckle as
opposed to deform plastically if the simulation is performed this way, even when the critical buckling strain is
less than the yield strain. This type of arbitrary behavior
can also be observed in previous atomistic studies of nanostructure buckling [47–54], but no explanation of, or remedy to, the problem has previously been given.
The reason behind this increment dependence is the high
symmetries of the systems which lead to unstable states of
equilibria and small potential energy gradients. Due to the
small potential energy gradients, the minimization algorithm may not be able to leave the unstable straight-beam
equilibrium configuration in favor of the more stable minimum energy state with bowing deformations. Hence the
obtained transition does not necessarily follow the minimum potential energy path but is rather governed by
chance. To overcome this complication, a perturbation
was employed to break the symmetry, and this was done
by perturbing the potential energy gradient by adding small
additional external forces acting on the free atoms in region
B so that the resulting force on the free atoms is written as:
Fi ¼ @U
þ dF;
@ri
ð4Þ
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d = 0.05 Å and d = 0.1 Å; however, when d = 0.2 Å, the
nanowire deformed plastically (cf. Fig. 3). When we did
perturb the system (dp = 0.1 Å) the instability occurred at
a lower strain and, unlike for the unperturbed simulations,
the obtained force–strain curve is not discontinuous. Even
though we have only reported results from one simulation
where a perturbation has been employed in this evaluation,
several simulations were performed with different strain
increment sizes and the resulting buckling properties from
those simulations did not show any strain increment size
dependence.
We have also compared the energy–strain curves for
perturbed and unperturbed systems (cf. Fig. 2b). As can
be seen, the energy–strain curves for the unperturbed systems are discontinuous in the same manner as for the
force–strain curves, whereas the curve of the perturbed system is continuous. In fact, it is seen that the energy curve
which corresponds to the lowest-energy transition path is
the curve of the perturbed system. Hence, that path is the
most likely transition path to occur, supporting the conjecture that the symmetry needs to be broken in order for the
lowest-energy transition to occur. From Fig. 2a it can be
seen that the force curve becomes almost perfectly horizontal at the critical point. The axial stress is related to the
potential energy curve through:
r¼
Fig. 2. Illustration of increment dependence of (a) the critical force and
(b) potential energy for perturbed and unperturbed systems as function of
strain. The perturbed system is denoted dp and the unperturbed systems
are denoted d. (c) Perturbation magnitude convergence study. (For
interpretation to colours in this figure, the reader is referred to the web
version of this paper.)
where U denotes the total potential energy and the vector
dF = (0, dF, dF) is the perturbation which is employed to
disturb the symmetry of the system. This means that the
minimization algorithm will find a slightly different state
of equilibrium than what would be found if we minimize
the potential energy. However, if we choose dF small enough, as we will discuss further in Section 3.3, this difference is negligible.
3.3. Numerical evaluation
In Fig. 2 we have illustrated the resulting differences
between simulations with and without employment of perturbations and also the influence of the perturbation size
on the force–strain behavior. In Fig. 2a we have compared
the force–strain curves for perturbed and unperturbed systems, and, as previously mentioned, for the unperturbed
systems we found that the type of instability which occurs
and the critical point at which instability occurs are highly
dependent on the strain increment size. In fact, the unperturbed systems buckled when the increment sizes were
F
1 dU
¼
;
A V d
ð5Þ
which means that when the force curve becomes horizontal, the potential energy curve becomes a linear function
of the strain. Thus, the energy curve transitions from a parabolic function to a linear function at the critical point,
which can be confirmed by inspection of Fig. 2b.
Moreover, in Fig. 2c we have performed a perturbation
convergence study to monitor the influence of the perturbation magnitude on the buckling properties. In the production simulations the perturbations were chosen to be of the
order 1011 < dF < 109 eV/Å, which were found to be sufficiently large to get the nanowires to buckle and small
enough not to influence the response notably. The sizes
of these perturbations are very small even in the context
of atomistic simulations, up to roughly 100 times the convergence criterion we utilized, where the root-mean-square
convergence tolerance for the system force was set to be
less than 1011 eV/Å. We monitored the total potential
energy of the nanowire and the potential energy of the individual atoms both for perturbed and unperturbed systems
up to the critical point, and the differences were found to be
of the order of 0.1–1 ppm. This shows that the influence of
the perturbation on the mechanical properties of the surfaces and the bulk is negligible. Nevertheless, they are
essential in order to get the nanowires to buckle at the critical point.
Even though some of the considered perturbations in
Fig. 2c are much exaggerated, larger than roughly 100
times the perturbations that were used in the production
runs, they illustrate how the transition character changes
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Fig. 3. Nanowires subjected to different strain increments after instability has occurred for (a) dp = 0.1 Å, (b) d = 0.05 Å, (c) d = 0.1 Å and (d) d = 0.2 Å.
The atoms are characterized using the method proposed by Ackland and Jones [59] (green = fcc, red = hcp, gray = unknown). The figures are generated
using AtomEye software [60]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. (a) The deflection of the midpoint of the nanowire cross-section as
a function of the compressive axial force and (b) the deflection of the
midpoint as a function of the applied axial strain. (c) Comparison of
straining from two different initial conditions, the solid line corresponds to
straining from perfect lattice distance and the markers correspond to
straining from the relaxed state.
when the nanowire is subjected to a perturbation that is too
large; even though the critical force remains essentially
unchanged it becomes harder to distinguish any distinct
point of instability. Thus, it becomes more difficult to
extract the critical strain with high accuracy when the perturbation is too large.
In Figs. 4a and b, we have compared the deflection of
the midpoint of a nanowire as functions of the axial
force and axial strain, respectively. From Fig. 4b it can
be seen that increasing the compressive axial strain does
not result in any buckling-type deflections until a critical
strain is reached. Once the critical strain is reached, the
deflections increase significantly. Furthermore, from
Fig. 4a it is observed that the axial force reaches a
threshold value after which the axial force does not
increase even if the axial strain is increased. However,
if the axial strain is increased, the deflection is also
increased. These observations are confirmations that it
is indeed buckling that occurs and that the numerical
scheme presented is appropriate for finding the critical
strains and loads.
Finally, in Fig. 4c we have investigated if the buckling
properties depend on which state we start to compress
the nanowire from. We did compress nanowires from the
relaxed state, where the nanowires have been allowed to
contract to balance the surface stresses, and from the ideal
state where the initial interplanar distance along the wire
axis corresponds to the ideal; note that Fig. 4c shows that
the compressive relaxation strain due to surface stress
effects (i.e. in the absence of any external loading) for the
particular nanowire considered in Fig. 4 is about 1.62%.
From Fig. 4c it can be seen that these two curves coincide
and thus the buckling properties do not depend on which
of the two initial configurations the nanowires are loaded
from. Consistently throughout this paper, we consider the
relaxed state to be the initial configuration.
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4. Results and discussion
4.1. Critical buckling forces
We have adopted two different approaches when evaluating the critical forces. First, we have compared critical forces
of different cross-sectional dimensions with the inverse
square aspect ratio. The reason for this can be deduced from
Eq. (1), where it is seen that the critical buckling force is proportional to the inverse square length of the nanowire. Thus,
in Fig. 5 we have plotted the critical load for different crosssections against the inverse square aspect ratio to study how
well this behavior is satisfied. Second, we have also studied
the normalized critical buckling forces (cf. Fig. 6). Here we
have made the critical force dimensionless, by instead considering the quantity jFcrjL2/(4p2EbI), where Eb is the
Young’s modulus of the bulk, which has values of 46.1 and
87.9 GPa for the h1 0 0i and h1 1 0i directions, respectively,
I is the moment of inertia, and jFcrj is the magnitude of the
Fig. 6. Normalized critical buckling force as functions of the aspect ratio
for (a) h1 0 0i/{1 0 0}, (b) h1 0 0i/{1 1 0} and (c) h1 1 0i/{1 1 0}{1 0 0} systems.
(For interpretation to colours in this figure, the reader is referred to the
web version of this paper.)
Fig. 5. The critical buckling force vs. the inverse square aspect ratio for (a)
h1 0 0i/{1 0 0}, (b) h1 0 0i/{1 1 0} and (c) h1 1 0i/{1 1 0}{1 0 0} systems. The
dashed lines are perfectly linear curves through the origin and the point
with the lowest value of (h/L)2 for the corresponding cross-section to
distinguish any deviations from the ideal inverse square length dependence. (For interpretation to colours in this figure, the reader is referred to
the web version of this paper.)
critical force measured from the MS simulations. This
dimensionless quantity is the same as the ratio between the
obtained critical force and that of the bulk, and sheds light
on other important issues such as elastic cross-sectional size
and aspect ratio dependencies.
From Fig. 5 it can be seen that the curves are quite close
to linear going through the origin. This is of course what is
expected from Eq. (1). Although it is hard to distinguish
any deviations of the curves in Fig. 5 from linearity going
through the origin, it appears that for nanowires with small
aspect ratios, i.e. when (h/L)2 is large, the deviations from
perfectly linear curves through the origin increase. This
observation can be confirmed from Fig. 6, where it is seen
that the curves are not perfectly horizontal which reveals
an unexpected aspect ratio dependence. It is shown that
the critical buckling force decreases for h1 0 0i/{1 0 0} and
h1 0 0i/{1 1 0} nanowires, while it increases with decreasing
aspect ratio for the h1 1 0i/{1 1 0}{1 0 0} nanowires. Moreover, it can be seen that the curves in Fig. 6 are not overlapping and this is an indication that there is a crosssectional size dependence in the flexural rigidity. We will
address these two issues separately.
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4.1.1. Cross-sectional size dependence
It is well known that the introduction of surfaces and
axial relaxations leads to elastic properties that deviate
from those of the bulk [45,61,62]. In particular, it has been
found that the Young’s modulus decreases with decreasing
cross-sectional size for nanowires when the axial direction
corresponds to a h1 0 0i direction, whereas for h1 1 0i directions it has been found that the Young’s modulus increases
with decreasing cross-sectional size [45,61,62]. Because of
the close relation between flexural rigidity and Young’s
modulus through Eq. (2), if the Young’s modulus varies
with the cross-sectional size, this will also influence the critical buckling force. In Fig. 6 it is seen that the curves are
not overlapping but rather differ in terms of magnitude.
For h1 0 0i/{1 0 0} and h1 0 0i/{1 1 0} nanowires the normalized critical forces are less than 1 for all the considered
nanowires and decrease with decreasing cross-sectional
size, whereas for h1 1 0i/{1 1 0}{1 0 0} nanowires we find
that the normalized critical forces are larger than what is
predicted by the bulk and are increasing with decreasing
cross-sectional size. Hence, the overall trends of the
cross-sectional size dependence observed for Young’s modulus by previous researchers are in accordance with the
observed cross-sectional size dependence of the normalized
critical forces in this work.
When comparing the curves in Figs. 6a and b it should
be noted that both of the considered systems have the same
h1 0 0i axial orientation but different types of bounding surfaces, i.e. {1 0 0} and {1 1 0} surfaces, respectively, which
makes it possible to study if and how the surfaces influence
the critical buckling forces. Comparing the systems with
the cross-section size 2.45 2.45 nm2 it can be seen
that the critical forces for the h1 0 0i/{1 0 0} system are
between 50% and 60% of the corresponding bulk value,
while for the h1 0 0i/{1 1 0} system the critical forces fall
between 70% and 80% of the corresponding bulk value.
Hence, the critical forces for the h1 0 0i/{1 1 0} system are
clearly higher than for the h1 0 0i/{1 0 0} system, which is
a general observation for all the considered cross-sections
in Figs. 6a and b.
We did extract the Young’s modulus for both the h1 0 0i/
{1 0 0} and h1 0 0i/{1 1 0} systems for the nanowires with the
cross-section size 2.45 2.45 nm2 from the linear elastic
regime of the stress–strain curves and they were found to
be 31.5 and 42.1 GPa, respectively. Since the bulk value
is 46.1 GPa, this means that the presence of {1 0 0} surfaces
leads to a more compliant structure than the {1 1 0} surfaces, which can also be observed by comparison of
Figs. 6a and b. The ratios between these finite-sized values
of the Young’s modulus and the bulk value are 68% and
91% for the h1 0 0i/{1 0 0} and h1 0 0i/{1 1 0} systems,
respectively. However, these ratios are higher than the normalized critical forces which were in the range 50–60% and
70–80%, respectively, which means that the cross-sectional
size dependencies of the normalized critical forces can only
partly be attributed to the cross-sectional size dependencies
of the Young’s modulus for the h1 0 0i/{1 0 0} and h1 0 0i/
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{1 1 0} systems. This observation was found to be true for
all the cross-sectional sizes.
For the h1 1 0i/{1 1 0}{1 0 0} system, extracting the
Young’s modulus from the linear elastic regime yielded a
value of 100.1 GPa for nanowires with a 3.32 3.47 nm2
cross-section. The ratio between this value and the corresponding bulk value, 87.9 GPa, is 1.14, which is in quite
good agreement with the normalized critical forces of
1.16 and 1.11 for nanowires with aspect ratios 20 and 30,
respectively. However, there are increasing discrepancies
when comparing with nanowires with greater aspect ratios
as it can be seen that the normalized critical forces are less
than 1.1 in the limit of large aspect ratios. It can thus be
concluded that in the limit of large aspect ratios, the normalized critical forces are smaller than the ratio between
the cross-sectional size-dependent Young’s modulus and
the Young’s modulus of the bulk for all crystallographic
orientations. We will address this further in Section 4.3.
Comparing the different systems, it is found that the deviations are smaller than for the h1 0 0i/{1 0 0} and h1 0 0i/
{1 1 0} systems, which implies that a large part of the
cross-sectional size dependencies of the normalized critical
forces for the h1 1 0i/{1 1 0}{1 0 0} systems can be attributed
to the cross-sectional size dependence of Young’s modulus,
which is in contrast to what was observed for the h1 0 0i/
{1 0 0} and h1 0 0i/{1 1 0} systems.
4.1.2. Aspect ratio dependence
Given the aforementioned cross-sectional size dependence of the Young’s modulus, theoretically all the curves
in Fig. 6 should be perfectly horizontal. The reason behind
this is that the ratio jFcrjL2/(4p2EbI) is merely the ratio
between the flexural rigidity of the nanowire and that of
the ideal bulk. The flexural rigidity is a cross-sectional
property and depends on the cross-sectional geometry
and the Young’s modulus, and we do not expect the
Young’s modulus to vary with the nanowire length. Thus
it is expected that the flexural rigidity will be constant for
a specific cross-sectional size with the aspect ratios that
we have considered in this paper, and that the curves in
Fig. 6 should be perfectly horizontal. As can be seen, there
are some deviations in the normalized critical force from
being perfectly horizontal. This implies that there is a slight
aspect ratio dependence which is not accounted for by Eq.
(1). It is also observed that the normalized critical forces
for the specific cross-sections do appear to approach an
asymptotic limit value as the aspect ratio increases. Nevertheless, on most accounts these deviations are quite small
and it appears as if the curves in Fig. 6 do not deviate significantly from horizontal curves and the variations
between aspect ratios 20 and 50 are less than 7%. Thus,
for the studied nanowires in this work, this deviation is
not as important as the cross-sectional size dependence of
the Young’s modulus in causing deviations of the critical
buckling force from the expected bulk value.
Although it is unlikely that shearing will have any significant influence on the buckling properties of nanowires
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Fig. 7. The solid lines are stress–strain curves for (a) h1 0 0i/{1 0 0}, (b)
h1 0 0i/{1 1 0}, and (c) h1 1 0i/{1 1 0}{1 0 0} systems. The dashed curves
corresponds to the tangential stiffness of the stress–strain curves just
before buckling occurs. (For interpretation to colours in this figure, the
reader is referred to the web version of this paper.)
with aspect ratios greater or equal to 20, we have employed
Timoshenko beam theory [63] to see whether this is an
effect due to shearing. It turned out that the influence
was minimal and certainly much smaller than the deviations from perfectly horizontal curves that can be seen in
Fig. 6, and so does not explain why the curves are not
horizontal.
If we study the separate figures, i.e. Figs. 6a–c, it can be
seen that the curves in each part seem to follow the same
trends and appear to look very similar except for the fact
that they are translated along the y-axis. What is also interesting is that the greatest deviation seems to be attained for
nanowires with the smallest aspect ratios. Taking Eq. (3) as
the starting point, it can be deduced that the critical strain
for systems with square cross-sections is given by
(p2/3)(h/L)2. Therefore, ideally, the critical strains will
be 0.82%, 0.36%, 0.21% and 0.13% for nanowires
with aspect ratios 20, 30, 40 and 50, respectively. It is therefore clear that larger compressive strains are needed for
smaller aspect ratio nanowires in order to reach the point
of buckling, which implies that nonlinear elastic contribu-
tions appear to become increasingly important for nanowires with small aspect ratios.
To investigate this in more detail, we studied stress–
strain curves to see whether there are any variations from
perfectly linear elastic behavior. In addition to the stress–
strain curves, we have also plotted the linear elastic tangent
stiffness curves from immediately before the instability
occurs. Fig. 7 reveals that there is a slight influence of nonlinear contributions for all nanowires and these influences
appear to be greatest for nanowires with an aspect ratio
of 20, despite the compressive buckling strains being less
than 1%. Furthermore, it can be seen that the tangent stiffness increases with increasing amounts of compressive
strain for the h1 1 0i/{1 1 0}{1 0 0} systems, while it
decreases for the h1 0 0i/{1 0 0} and h1 0 0i/{1 1 0} systems.
This is exactly what happens in Figs. 5 and 6 as the aspect
ratio decreases. Moreover, the magnitude of the differences
between the tangent modulus before buckling and the
Young’s modulus is greatest for the h1 1 0i/{1 1 0}{1 0 0}
system, which is also in accordance with the observations
from Fig. 6c. For the h1 1 0i/{1 1 0}{1 0 0} system, the tangent modulus immediately before buckling was found to
be roughly 13% higher than the Young’s modulus for the
nanowire with an aspect ratio of 20, which means that
the normalized critical force for the aspect ratio 20 nanowire should lie somewhere between 0% and 13% higher
than for the aspect ratio 50 nanowire. Inspection of
Fig. 6c reveals that the normalized critical force value for
the aspect ratio 20 nanowire falls within that range. For
the h1 0 0i/{1 0 0} and h1 0 0i/{1 1 0} systems we find
that the tangent moduli immediately before buckling were
8% and 6% smaller than the Young’s modulus, respectively. Thus, we expect that the maximum deviations in
Figs. 6a and b are less than 8% and 6% lower than the
asymptotic value, respectively, which also can be confirmed
by inspection.
This nonlinear influence also explains why the normalized critical force for a specific cross-sectional size and crystallographic orientation approaches an asymptotic limit
value as the aspect ratio increases. This is because the critical strain decreases with increasing aspect ratio and nanowires with large aspect ratios are therefore not influenced
by nonlinear elastic properties to the extent that nanowires
with small aspect ratios are. Clearly, there is a strong correlation between the deviations from perfectly horizontal
behavior in Fig. 6 and the nonlinear elastic behavior in
Fig. 7. Hence, it appears to be very likely that the variations in Figs. 5 and 6 for the critical buckling force for a
given cross-sectional size are due to a nonlinear elastic contribution to the stiffness. Thus, even though we will not
address it here, it is likely that this aspect ratio dependence
of the critical force can be systematically modeled by an
elastic continuum model.
4.1.3. Wang and Feng buckling model
In addition to comparisons with classical beam theory,
we have also compared our results with the buckling model
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P.A.T. Olsson, H.S. Park / Acta Materialia 59 (2011) 3883–3894
Fig. 8. Comparison of the normalized critical buckling forces from MS
simulations and the Wang and Feng model for h1 0 0i/{1 0 0} nanowires
with (a) s0 = 1.33 J/m2 and (b) s0 = 0. The solid lines correspond to results
from the Wang and Feng model and the markers correspond to MS
results. (For interpretation to colours in this figure, the reader is referred
to the web version of this paper.)
recently proposed by Wang and Feng (WF) [55]. This comparison is of interest because the WF model is based upon
the linear surface elasticity theory first proposed by Gurtin
and Murdoch [64], in which the surface is assumed be rigidly attached to the bulk, and have its own elastic properties through the following relationship:
s ¼ s0 þ Es ;
ð6Þ
where s is the surface stress, s0 is the residual (strain-independent) surface stress, Es is the surface stiffness and Es is
the strain-dependent surface stress. The various material
constants for h1 0 0i/{1 0 0} systems were obtained from
our atomistic model, and were found to be s0 = 1.33 J/m2
and Es = 4.93 J/m2 and Young’s modulus for the bulk
Eb = 46.1 GPa.
Using the WF model for the h1 0 0i/{1 0 0} systems considered here, with square cross-sections and biclamped
boundary conditions, the normalized critical force
becomes:
2
jF cr jL2
8Es
6 s0
L
¼
1
þ
þ
:
ð7Þ
2
2
4p Eb I
E b h p Eb h h
In Fig. 8a we have plotted the normalized critical forces
from MS simulations and those predicted by the WF model
as a function of the aspect ratio. The WF model predicts
significant increases in the critical force due to surface effects, up to about 20 times the critical force predicted by
the bulk, whereas the MS simulations predict that the critical forces are smaller than those predicted by the bulk. The
decrease in normalized critical forces is found to be less
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than 50% for the MS simulations, significantly different
from the increase of the order of 200–2000% predicted by
the WF model. For example, for the 2.45 2.45 nm2 nanowires, the WF model predicts that critical forces are 3.5 and
19 times larger than the corresponding bulk values when
the aspect ratios are 20 and 50, respectively. This can be
compared with the MS results that lie between 50% and
60% of the corresponding bulk value. This suggests that
the WF model does not accurately capture the critical
buckling forces of nanowires due to surface effects. In particular, from Fig. 8a it can be seen that the discrepancies
between the WF model and the MS results increase in a
parabolic manner as the aspect ratio increases. This implies
that the last term on the right-hand side of Eq. (7), which is
the only term that varies with the aspect ratio, accounts for
most of the diverging discrepancies between MS results and
WF calculations.
To investigate this in more detail we assume that the last
term on the right-hand side of Eq. (7) is zero, i.e. s0 = 0, to
see how well the MS results and the WF calculations agree
in such circumstances. In Fig. 8b it can be seen that the
agreement between MS results and the WF model is significantly improved when s0 = 0. This is in line with observations from other researchers where the eigenfrequency
spectrum of silicon nitride cantilevers has been found to
be in good agreement with experimental results if s0 = 0
[65]. The remaining terms of the right-hand side of Eq.
(7) merely constitute the ratio between the size-dependent
flexural rigidity and that of the bulk, which implies that
the critical force can be modeled through a surface elastic
model. It should, however, be noted that a linear elastic
uniaxial model in general may be too simple to capture
accurately the critical properties. Clearly, it does not capture the nonlinear behavior in Figs. 6 and 7 or any nonlinear relaxation effects, hence it may be necessary to consider
this problem with a more general three-dimensional nonlinear elastic surface model.
4.2. Critical buckling strains
In addition to the normalized critical force, we have also
studied the equivalent normalized strain, jcrjAL2/(4p2I),
where jcrj is the magnitude of the critical strain. Ideally
these curves should be horizontal; they should also overlap
as the Young’s modulus has been eliminated from the
equation, and thus all these ratios should be unity. In the
same manner as for the critical forces, we also find some
variations among the normalized strains (cf. Fig. 9). Some
of these variations can be attributed to the perturbation
load as it may influence the critical strains somewhat as discussed in Section 3.3. The normalized critical strains are
also much more sensitive to variations as they are very
small, even for the bulk. Nevertheless, one can distinguish
some clear trends. As for the normalized critical forces, it
can be seen that the normalized strain curves do not overlap. It is observed that the normalized critical strain
decreases with decreasing cross-sectional dimensions. The
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P.A.T. Olsson, H.S. Park / Acta Materialia 59 (2011) 3883–3894
Fig. 9. Normalized critical buckling strain as functions of the aspect ratio
for (a) [1 0 0]/{1 0 0}, (b) [1 0 0]/{1 1 0} and (c) [1 1 0]/{1 1 0}{1 0 0} systems.
(For interpretation to colours in this figure, the reader is referred to the
web version of this paper.)
normalized critical strains are closer to unity than the normalized critical forces and the deviations from unity are up
to 20%, whereas for the normalized critical forces they are
up to 50%.
In Figs. 9a and b we have plotted the normalized critical
strains for the h1 0 0i/{1 0 0} and h1 0 0i/{1 1 0} systems,
respectively. The magnitudes of the normalized critical
strains for the h1 0 0i/{1 0 0} and h1 0 0i/{1 1 0} systems are
roughly the same, despite the differences observed for the
normalized critical force. Comparing the magnitudes of
the normalized critical strains for the h1 0 0i/{1 0 0} and
h1 0 0i/{1 1 0} systems with those of the h1 1 0i/
{1 1 0}{1 0 0} system in Fig. 9c, it can be seen that the
strains for the h1 1 0i/{1 1 0}{1 0 0} system are much closer
to unity.
One possible explanation as to why the different curves
do not overlap and why they differ from unity is the simplification of the flexural rigidity. Reducing Eq. (1) to Eq. (3)
requires that the flexural rigidity in Eq. (1) can be written
as EI, where E and I denote the Young’s modulus and
moment of inertia, respectively. However, this is not necessarily accurate because the actual definition of the flexural
rigidity is given by Eq. (2), which means that in order for
(EI) = EI to be valid, the Young’s modulus must be constant, or homogeneous, over the entire cross-section. However, it has been demonstrated that surfaces have elastic
properties that are different to those of the underlying bulk
[45,61], which leads to a nanowire cross-section with varying elastic properties. This means that the surfaces that
undergo the greatest amounts of strain during bending
and have the greatest lever arm from the neutral axis have
different elastic properties than the bulk.
It can also be seen that all the critical buckling strains
are smaller than their bulk counterparts, which indicates
that the surfaces are more compliant than the bulk, which
is in agreement with results from previous researchers [45].
Thus it can be expected that the bending moment is overestimated and not accurately described this way, which
would explain why the curves in Fig. 9 are not overlapping.
The critical strains do nevertheless converge towards those
predicted for the bulk when the cross-sectional dimensions
increase. The fact that all the obtained strains are smaller
than what is predicted by bulk calculations implies that
the validity of using Eq. (1) to extract the Young’s modulus
may not in practice be accurate. Clearly, if the strains are
not accurately predicted by Eq. (3), this is a clear indication
that (EI) – EI and that complicates the extraction of the
Young’s modulus as it cannot be considered a uniform,
homogeneous and constant quantity.
We also note that in previous publications we were able
to describe the eigenfrequency spectrum of [1 0 0]/{1 0 0}
gold nanowires very accurately without the use of any
explicit surface elastic models, which is in contrast to the
findings in this paper [35,36]. There are, however, two possible explanations that can shed light on these inconsistencies. First, the considered nanowire sizes in Refs. [35,36] are
4.08 4.08 nm2 and larger, and the variations in mechanical properties increases quite nonlinearly for smaller nanowires. Thus, we expect larger variations for smaller
nanowires. Second, when plane transverse eigenfrequencies
of unstressed beams are calculated, these are found to be
proportional
pffiffiffiffiffiffiffiffiffiffiffi the square root of the flexural rigidity, i.e.
fi ¼ k i ðEIÞ [66]. This means that any minor variations
in buckling properties due to variations in flexural rigidity
will result in even smaller variations when the eigenfrequency spectrum is considered.
4.3. Coupling between normalized critical forces and strains
The observation that the normalized critical strains are
smaller than unity may be the explanation for the finding
in Section 4.1.1, where it was found that the normalized
critical forces were lower than the ratio between the
finite-sized Young’s modulus and the Young’s modulus
of the bulk in the limit of large aspect ratios. In fact, there
appears to be a correlation between the normalized critical
strains, the discrepancies between the ratio of Young’s
modulus of nanowires and the bulk and the normalized
critical forces in the limit of large aspect ratios.
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P.A.T. Olsson, H.S. Park / Acta Materialia 59 (2011) 3883–3894
To illustrate this we take the 4.08 4.08 nm2 nanowire of the h1 0 0i/{1 0 0} system with an aspect ratio of
50. Close inspections of Figs. 6 and 9 reveal that the
normalized critical force and strain are 0.77 and 0.95,
respectively, and the ratio between the finite-sized
Young’s modulus and that of the bulk is 0.83. Thus
we obtain: 0.830.95 = 0.79 0.77. We performed this
type of analysis for nanowires with an aspect ratio of
50 for all the systems and cross-sectional sizes and found
that the deviations were in general less than 3%. To
describe this in words: (i) we obtain a deviation in the
critical strains which we believe is due to the compliant
surfaces; and (ii) we have a size-dependent Young’s modulus. Together these two discrepancies quite accurately
account for the deviations between the normalized critical forces and the ratio between finite-sized and bulk
Young’s modulus in the limit of large aspect ratios. This
is due to the fact that the buckling strains are small for
nanowires with large aspect ratios and the influence of
nonlinear elastic properties are therefore small. This
can be confirmed by inspection of Fig. 7, and shows that
the relation between the critical strain and the critical
force is close to linear elastic for nanowires with high
aspect ratios, where the coefficient of proportion is the
size-dependent Young’s modulus.
5. Summary and conclusions
In this paper we have studied surface and axial orientation effects on the buckling properties of gold nanowires
through MS simulations. As a part of the work we have
developed a simple numerical scheme which utilizes small
perturbations of the potential energy gradients. This
scheme overcomes previously observed numerical complications such as discontinuous stress–strain and energy–
strain curves and strain increment size dependencies. It
accurately describes the critical forces and strains, and produces an accurate energy transition curve between the initial state and the buckled state. The key findings of this
work are:
1. The normalized critical buckling forces are dependent
on the axial orientation of the nanowires, where the
normalized buckling forces decrease for both the
h1 0 0i/{1 0 0} and h1 0 0i/{1 1 0} systems, whereas they
increase with decreasing cross-sectional size for the
h1 1 0i/{1 1 0}{1 0 0} system. The critical buckling forces
are also dependent on the surface orientation where
the h1 0 0i wires with {1 0 0} surfaces had around 820% lower critical buckling forces than the h1 0 0i
wires with {1 1 0} surfaces depending on the cross-sectional size.
2. The normalized critical buckling forces were found to
show deviations from the expected inverse square length
dependence for smaller aspect ratio nanowires. This is
due to the fact that smaller aspect ratio nanowires
require larger strains to induce buckling, which makes
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the influence of nonlinear elastic properties more important and results in increased and reduced stiffness for
h1 1 0i and h1 0 0i wires, respectively. Overall, this nonlinear elastic influence in stiffness is not as important
as the cross-sectional size-dependent Young’s modulus
in causing deviations of the critical buckling force from
the expected bulk value.
3. For the h1 0 0i/{1 0 0} system, we compared our MS
results for the critical buckling load with a continuum
surface elasticity model recently proposed by Wang
and Feng [55]. In contrast to the MS results, the
WF model predicted that the critical forces, including
nanoscale surface effects, should increase significantly
as compared to the critical bulk buckling force, which
is in contrast to the MS results which showed that the
critical forces decreased compared to the corresponding critical forces of the bulk material.
4. It was found that all the critical buckling strains are
lower than their bulk counterparts and that the magnitudes of the critical strains decrease with decreasing
cross-sectional dimensions for all surface and axial orientations that were considered. The smaller critical
buckling strains are most likely due to the heterogeneous
character of the cross-section due to the fact that the
surfaces are more compliant than the bulk, which means
that these variations must be taken into account when
calculating the flexural rigidity. This suggests that the
accuracy of Eq. (3) may serve as an indicator as to the
validity of using continuum mechanics relationships to
extract the Young’s modulus from buckling experiments
or simulations.
5. Finally, we found a linear elastic correlation between the
normalized critical strains and forces for nanowires with
high aspect ratios. This suggests that the critical force is
close to linearly proportional to the critical strain, and
that the constant of proportion is the size-dependent
Young’s modulus.
Acknowledgments
P.A.T.O. gratefully acknowledges the funding provided by the Swedish research council. H.S.P. gratefully
acknowledges the support of NSF Grant 0750395. The
simulations were performed using the computational resources at LUNARC, Center for Scientific and Technical
Computing, Lund University.
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